# Calculating the residue of a complex function

1. Apr 12, 2012

### AlBell

1. The problem statement, all variables and given/known data

calculate the residue of the pole at z=i of the function

f(z)=(1+z^2)^-3

State the order of the pole

2. Relevant equations

I know the residue theorem and also the laurent series expansion but I'm having trouble applying these

3. The attempt at a solution

I think the pole is order 3 but I'm having trouble either expanding the function in order to find the a(-1) term or implementing the residue theorem and would appreciate some help! Thanks

2. Apr 12, 2012

### micromass

Staff Emeritus
Do you know that the residue of a function f at the point a can be calculated by

$$\lim_{z\rightarrow a} (z-a)f(z)$$

3. Apr 12, 2012

### AlBell

I thought it was ?

4. Apr 12, 2012

### micromass

Staff Emeritus
Yes, of course, sorry. My formula is for a simple pole.
So, can you use your formula to find the residue?

5. Apr 12, 2012

### AlBell

Yes but I am having trouble actually implementing the formula and wondered if anyone could do an example to help me understand it?

6. Apr 12, 2012

### micromass

Staff Emeritus
Well, first you'll need to determine the order of the pole.

Your guess is that the order of the pole is 3. So you'll need to look at

$\lim_{z\rightarrow a} (z-a)^3f(z)$.

This limit should be nonzero for the order of the pole to be 3. If it is zero, then the pole is of a lower order. If the limit doesn't exist, then the pole is of higher order.

So, can you calculate that limit?

Hint: factor the denominator.

7. Apr 13, 2012

### AlBell

Ah I forgot about factorising, that makes it so much simpler! I get 3/8 I think!

8. Apr 13, 2012

### Robert1986

VERY close. You probably got something like 12*(2i)^(-5), right? Just compute this again. The 3/8 is right, but you are missing two factors.